Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures

Hamish Flynn, David Reeb, Melih Kandemir, Jan Peters

Keywords: Linear bandits, confidence sequences, martingales, convex optimization, cumulative regret, regret analysis

TL;DR: Based on novel mixture martingales, we obtain tighter confidence bounds for linear bandits resulting in better algorithms with performance guarantees.

Abstract: We present improved algorithms with worst-case regret guarantees for the stochastic linear bandit problem. The widely used "optimism in the face of uncertainty" principle reduces a stochastic bandit problem to the construction of a confidence sequence for the unknown reward function. The performance of the resulting bandit algorithm depends on the size of the confidence sequence, with smaller confidence sets yielding better empirical performance and stronger regret guarantees. In this work, we use a novel tail bound for adaptive martingale mixtures to construct confidence sequences which are suitable for stochastic bandits. These confidence sequences allow for efficient action selection via convex programming. We prove that a linear bandit algorithm based on our confidence sequences is guaranteed to achieve competitive worst-case regret. We show that our confidence sequences are tighter than competitors, both empirically and theoretically. Finally, we demonstrate that our tighter confidence sequences give improved performance in several hyperparameter tuning tasks.

Submission Number: 14987